How to interpret couplings in optimal transport? Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to
$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable mappings }T: X \rightarrow Y \text{ and } T_\#\mu = \nu$$
has a relatively straightforward interpretation: We try to find a measurable map $T$ that minimizes the cost to move mass from $x$ to $T(x)$. Now, the Kantorovich problem to
$$ \text{minimize} \quad \int c(x,y)\pi(dx,dy) \quad \text{over couplings } \pi \text{ with first and second marginals } \mu \text{ and } \nu \text{, respectively,}$$
I find to be much harder to interpret as a real `mass transfer' problem; If $\pi^\star$ is an optimal coupling to the Kantorovich problem, what does $\pi^\star$ tell me where how much mass really goes? How do I interpret the Kantorovich problem?
 A: Of the mass $\mu(A)$ in $A$ a fraction $\pi(A \times B)$ is transported to $B$, so you can think of this as a randomized transport map. A basic example to think of is $\mu=\delta_0$ and $\nu=(\delta_1+\delta_{-1})/2$. Half the mass at 0 is sent to 1 and half is sent to -1. You can get a better intuition from reading more about construction of couplings:
[1] Lindvall, Torgny. Lectures on the coupling method. Courier Corporation, 2002.
[2] Thorisson, Hermann. "Coupling methods in probability theory." Scandinavian journal of statistics (1995): 159-182.
A: The answers above already say it all.
Still, if you are a visual person, have a look at the following optimal coupling/transport plan (or to be precise, a sample thereof. Note that the colour's strength is a function of the empirical density.) between a univariate $T$ distribution with 3 degrees of freedom and a standard Gaussian. You can really interpret this as "to reshape optimally a $T_3$ distribution to get a Gaussian, I need to push the mass contained in the tails towards the middle". To see this, observe that an infinitesimal volume element around -4 for the student will approximately be transported to -2. This is a particularity of the 1D case, but the intuition "holds" for higher dimensions.

A: Given any coupling $\pi$ between two measures $\mu$ and $\nu$, you can construct a
associated transport plan. To do so, for each point $x \in X$, you distribute the mass according the "marginal distribution" of $\pi$ at $x$. At first, this does not seemed well defined, since for non-atomic measures singletons will be sets of measure zero. However, there is a well defined disintegration of $\pi$, which gives a well-defined marginal $\mu$-almost everywhere, which is how to define the transport rigorously. In the picture below, I've tried to draw the "marginal of $\pi$ at $x$" in purple.

Kantorovich's theorem says that the optimal coupling is often concentrated on a much smaller subset of $X \times Y$, and a solution to the Monge problem transports each point $x \in X$ to a unique point $y \in Y$. However, the transport is defined for an arbitrary coupling.
